Coupling Constant

The coupling constant is denoted by \(\Gamma\) and it corresponds to

\[\Gamma = \frac{(Ze)^2}{4\pi \varepsilon_0 k_B T a_{\textrm{ws}}},\]

In the case of mixtures the sarkas.core.Parameters.calc_coupling_constant() method calculates the effective \(\Gamma\) as

\[\Gamma_{\textrm{eff}} = \sum_{\alpha} c_{\alpha} \Gamma_{\alpha}, \quad \Gamma_{\alpha} = \frac{(Z_{\alpha}e)^2}{4 \pi \varepsilon_0} \frac{1}{ k_B T a_{\alpha}},\]

where \(a_{\alpha}\) is Wigner-Seitz radius of species \(\alpha\) defined as

\[a_{\alpha} = a_{\textrm{ws}} \left ( \frac{Z_{\alpha}}{\left \langle Z \right \rangle} \right )^{1/3},\]

Note that we can rewrite the effective \(\Gamma\) as

\[\Gamma_{\textrm{eff}} = \left \langle Z^{5/3} \right \rangle \left \langle Z \right \rangle^{1/3} \frac{e^2}{4\pi \varepsilon_0} \frac{1}{k_B T a_{\textrm{ws} }},\]

which is the definition usually found in the literature.